On the fourth power moment of the Riemann zeta-function
The second author [Acta Math. (to appear)] has recently given a sharp formula for \[ \int_ \infty^ \infty \bigl | \zeta \bigl( {\textstyle {1\over 2}}+ iT+ it \bigr) \bigr|^ 4 e^{-t^ 2/ \Delta^ 2} dt, \] with a good error term, and additional terms involving Maass wave form \(L\)-functions. The present paper gives applications to the error term \[ E_ 2 (T)= \int_ 0^ T \bigl| \zeta \bigl( {\textstyle {1\over 2}}+ it \bigr) \bigr|^ 4 dt- TP_ 4 (\log T), \] in the usual asymptotic formula for the fourth power moment of the zeta-function. It is shown that \(E_ 2 (T)\ll T^{2/3} (\log T)^ A\) for an absolute constant \(A>0\), improving by a factor \(T^ \varepsilon\) a bound of \textit{N. I. Zavorotnyj} [Automorphic functions and number theory, I, Vladivostok, 69-125 (1989; Zbl 0711.11028)] and that \(E_ 2 (T)= \Omega (T^{1/2})\). Moreover \textit{H. Iwaniec}'s bound [Sémin. Théor. Nombres, Univ. Bordeaux, 1979/80, Exposé No. 18 (1980; Zbl 0438.10030)] for \[ \sum_{r\leq R} \int_{t_ r}^{t_ r +\Delta} \bigl| \zeta \bigl( {\textstyle {1\over 2}}+ it \bigr) \bigr|^ 4 dt \] is improved, by replacing a factor \(T^ \varepsilon\) by a power of \(\log T\).
scientific article; zbMATH DE number 749189
scientific article